I just can't get the jist of it. I can see what they look like on wolfram alpha, but I wont have access to it in say an exam situation and just can't seem to grasp it when trying to do them.
Are there any convenient ways of being able to quickly and sufficiently learn to do similar cos waves?

Yes you shrink the domain of study to draw part of the wave and then finish it off because the cosine function is an even function. ( f(x)=f(-x) ). You can also count the derivative and see where it equals zero; that means the tangent line is horizontal or parallel to the x-axis and you will get a convex or concave curve. In addition to that you can also count the derivative of other points to get their slope so it can make it easier for you to draw the diagram.